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Comment. Math. Helvetici 58 (1983) 529-542 0010-2571/83/040529-14501.50 + 0.20/0 1983 Birkh~iuser Verlag, Basel Group actions on fibered three-manifolds ALLAN L. EDMONDS 1 and CHARLES LIVINGSTON 1 1. Introduction In this paper we present several results about finite group actions on three- dimensional manifolds. The results are primarily directed toward a geometric understanding of periodic knots in the 3-sphere, that is, knots left invariant by periodic homeomorphism of S 3 which fix a simple closed curve in the complement of the knot. One noteworthy application of the present techniques is that nontrivial periodic knots have "Property R," that is, surgery on such a knot cannot produce SlxS 2. Let G be a finite group acting effectively and smoothly on an orientable 3-manifold M, preserving orientation. The first result is that if the orbit manifold M* contains an incompressible surface F*, then (after suitably adjusting the embedding) the preimage of F* in M is an incompressible surface. The proof of this makes use of the Equivariant Loop Theorem of Meeks and Yau [12, 13] and isrgiven in Section 2. An immediate corollary of this result is that a periodic knot has an invariant incompressible Seifert surface. If there is a bound on the possible genera of the incompressible Seifert surfaces of a given knot K (as is the case for fibered knots), then the Riemann-Hurwitz formula places nontrivial bounds on the possible periods of K. See Section 2. There is an extensive literature devoted to the problem of determining the possible periods of a given knot [1,5,6,7, 11, 15, 16, 18,22]. Most previous work on this problem has been heavily algebraic in nature, in contrast to the present more geometric approach. In Section 3 the preceding work is applied to prove that periodic knots have Property R. Now suppose that F is a compact, orientable surface and that G acts on F x [0, 1] preserving orientation and leaving both F x {0} and F x {1} invariant. We show that the action is equivalent to the level-preserving action which is the product of the action of F x {0} with the trivial action on the interval [0, 1], except Supported in part by grants from the National Science Foundation. 529

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Comment. Math. Helvetici 58 (1983) 529-542 0010-2571/83/040529-14501.50 + 0.20/0 �9 1983 Birkh~iuser Verlag, Basel

G r o u p ac t ions o n f ibered t h r e e - m a n i f o l d s

ALLAN L. EDMONDS 1 and CHARLES LIVINGSTON 1

1. Introduction

In this paper we present several results about finite group actions on three- dimensional manifolds. The results are primarily directed toward a geometric understanding of periodic knots in the 3-sphere, that is, knots left invariant by periodic homeomorphism of S 3 which fix a simple closed curve in the complement

of the knot. One noteworthy application of the present techniques is that nontrivial

periodic knots have "Proper ty R ," that is, surgery on such a knot cannot produce S l x S 2.

Let G be a finite group acting effectively and smoothly on an orientable 3-manifold M, preserving orientation. The first result is that if the orbit manifold M* contains an incompressible surface F*, then (after suitably adjusting the embedding) the preimage of F* in M is an incompressible surface. The proof of this makes use of the Equivariant Loop Theorem of Meeks and Yau [12, 13] and

isrgiven in Section 2. An immediate corollary of this result is that a periodic knot has an invariant

incompressible Seifert surface. If there is a bound on the possible genera of the incompressible Seifert surfaces of a given knot K (as is the case for fibered knots), then the Riemann-Hurwitz formula places nontrivial bounds on the possible periods of K. See Section 2. There is an extensive literature devoted to the problem of determining the possible periods of a given knot [ 1 , 5 , 6 , 7 , 11, 15, 16, 18,22]. Most previous work on this problem has been heavily algebraic in nature, in contrast to the present more geometric approach.

In Section 3 the preceding work is applied to prove that periodic knots have

Property R. Now suppose that F is a compact, orientable surface and that G acts on

F x [0, 1] preserving orientation and leaving both F x {0} and F x {1} invariant. We show that the action is equivalent to the level-preserving action which is the product of the action of F x {0} with the trivial action on the interval [0, 1], except

Supported in part by grants from the National Science Foundation.

529

530 A L L A N L. E D M O N D S A N D C H A R L E S LIVINGSTON

possibly when the action on F • {0} has fewer than four exceptional orbits and orbit space the 2-sphere. The proof is given in Section 4 and makes essential use of the Equivariant Dehn Lemma of Meeks and Yau [12, 13] and of the solution of the classical Smith Conjecture [19], which corresponds to the case here that F is a 2-disk. After the first version of this paper was written we learned that W. Meeks

and G. P. Scott have proved the above result in the more difficult excluded case as well.

An easy application of the preceding result, given in Section 5, is that quite generally a group action on a surface bundle over the circle with unique fibering is equivalent to a fiber-preserving action. In particular, any periodic fibered knot in S 3 admits a fibering preserved by the corresponding group action.

The final section contains further applications of the preceding results to group actions preserving fibered knots. In particular one has a very short list of fibered knots of a given genus g which admit periods m which are maximal (m ~ 2g + 1) or nearly maximal (m = g + 1). Several instructive examples are given as well. Finally we show that the present techniques give a partial answer to a question of D. Goldsmith in the Kirby problem set [9; 1.28].

The techniques used in this paper are similar to those used by Gordon and Litherland [4]. That work is primarily concerned with invariant incompressible surfaces in the complement of the exceptional set, while our main interest is in incompressible surfaces which intersect the exceptional set.

Quite similar and more complete results in the case G =7/2 are due to Tollefson [21] and Kim and Tollefson [8].

2. Lifting incompressible surfaces

Throughout this section let G be a finite group acting effectively and smoothly on an orientable, differentiable 3-manifold M preserving orientation. The excep- tional set E for the action is the set of all points in M having nontrivial isotropy group. If nonempty, E is a one-dimensional C W complex with all vertices of valence one lying in OM. The exceptional set is a 1-manifold precisely if all

isotropy groups are cyclic. Let M* denote the orbit manifold and let p : M---> M* be the orbit map. The set B = p(E) is called the branch set.

Two embedded surfaces F and F ' in a 3-manifold N are said to be disk

equivalent if there is a sequence Fz = F, F 2 . . . . . F n = F ' of surfaces in N such that for each k, 1 < k --< n, there exist disks Dk c i n t Fk and D~ c i n t Fk-1 such that

F k - D k = F k - 1 - D~. If N is irreducible, then standard techniques show that disk equivalent surfaces are isotopic.

Let F be a surface in a 3-manifold N such that F fq ON = OF. A compressing

Group actions on fibered three-manifolds 531

disk for F is a 2-disk D c i n t N such that D N F = OD and OD is a homotopicaUy nontrivial simple loop on F. If F has a compressing disk or is a nullhomotopic 2-sphere or is a 2-disk which can be deformed rel OF into ON, then F will be

called compressible; otherwise F is said to be incompressible. A surface F in the orbit manifold M* is said to be transverse to the branch set

B provided that F only meets B in the subset B0 of points where B is a 1-manifold and F is transverse to B0 in the usual sense.

As indicated previously we shall need the following basic result of Meeks-Yau [12, 13]. We quote from [4].

E Q U I V A R I A N T LOOP THEOREM. Let N be a 3-manifold, and G a finite group acting on N. Suppose F is a compressible component of ON. Then there exists a compressing disk D for F such that for all g ~ G, either g(D) = D or g(D) O D = r

Although Meeks and Yau only assert the result when N is compact and orientable, the theorem is true in the generality given. Further one may assume G

acts freely on G(OD).

T H E O R E M 2.1. Any two-sided incompressible surface in the orbit manifold M * is disk equivalent to an incompressible surface F* which meets the branch set B transversely and such that F = p-~(F*) is a two-sided incompressible surface in M.

Proof. It may be assumed that F* is an incompressible surface in M* which is transverse to B and meets the non-singular part of B in the minimum number of points possible for incompressible surfaces in its disk equivalence class. Transver-

sality implies that F = p-X(F*) is a surface in M. Suppose that F is compressible in M. If a component Fx of F is a null-

homotopic 2-sphere, then F 1 = OA where A is a homotopy 3-cell in M. Let H be the subgroup of G leaving A invariant. Then F=OH(A) and F*=Op(A), a 2-sphere. Moreover p(A) is a simply connected 3-manifold with boundary F*,

and this implies that F* is compressible. Similarly, if some component F~ of F is a disk homotopic into 0M, it follows

that F* is a disk homotopic into OM*, contradicting incompressibility of F*. Finally suppose F contains no spheres or disks. Let N be an invariant tubular

neighborhood of F in M. Then the manifold W = ( M - OM) - int N has as bound- ary two copies of int F, and has compressible boundary. By the Equivariant Loop

Theorem there is a compressing disk D1 for OW in W such that for any g ~ G either g(D~) = Dx or g(D1) n D1 = •. By using an equivariant product structure in N one may expand Da to a compressing disk D for F such that for g ~ G either

g ( D ) = D or g(D) N D = Q .

532 A L L A N L. E D M O N D S A N D C H A R L E S LIVINGSTON

Let D * = p(D) in M*. Then D * is a disk transverse to the branch set B, D * A B consists of at most one point and D*OF*=OD*. Let F~0 be the component of F* meeting D*. First suppose F* is neither a 2-sphere nor a 2-disk. Since FO* is incompressible there is a 2-disk Do* c F* with OD~ = 8D*. The minimality condition on F* implies that D~ meets B in at most one point. Therefore each component of p-~(Do*) is a disk in F. Since the boundary of some component of p-t(D*o) is OD, it follows that D is not a compressing disk for F after all. If F* is a 2-sphere, then OD* divides Fo* into two 2-disks FI* and F2*. Since F contains no 2-sphere and 0D is nontrivial on F, each F* must contain at least two branch points, while D* contains at most one. One of the two 2-spheres D* U F * or D* O F * must be incompressible and mee t B in fewer points than Fo* did. This contradicts the choice of F. If Fo* is a 2-disk, then 0D* divides Fo* into a 2-disk F~* and an annulus F2*. Since no component of F is a 2-disk, and OD is nontrivial on F, p-l(F*) does not consist of disks, and so F* meets B at least twice. But D* meets B at most once. Thus a disk move would reduce F * N B, a contradiction. This completes the proof. [ ]

Remark. It follows that if the orbit manifold M * is sufficiently large, then so

is M.

C O R O L L A R Y 2.2. Let K be a knot in an integral homology 3-sphere ,~ invariant under a semifree orientation-preserving action of the cyclic group Cm of order m, with fixed set A (the "axis") disjoint from K. Then K bounds an incompressible Seifert surface invariant under C~.

Proof. Since Cm has fixed points the quotient map p:2~ ~ 2~* induces a surjection H I ( ~ ; 7]) --~ Hx(~*; 7]). It follows that ~ * is also an integral homology 3-sphere containing the knot K * = p(K). Let F* be an incompressible Seifert surface for K* meeting the branch set transversely a minimum number of times. Then by Theorem 2.1 p-~(F*) is the required incompressible Seifert surface for

K. [ ]

Note that in the situation of Corollary 2.2 the linking number lk(K, A ) = lk(K*, A*) is relatively prime to m. Otherwise p-l(K*)=Op-l(F*) would be

disconnected. The following well known lemma is the basis for further restrictions on the

periods of knots.

L E M M A 2.3. Let F be a compact, connected, orientable surface of genus g > 0 with one boundary component. If F admits a semifree, orientation-preserving action of the cyclic group C,, with nonempty fixed set, then m <- 2g + 1.

Group actions on fibered three-manifolds 533

Proof. The Riemann-Hurwi tz formula for the regular branched cyclic covering p : F ~ F* says that

1 - 2 g = m(1 - 2 g * ) - k ( m - 1)

where g* is the genus of the orbit surface F* and k is the number of branch points. If k = l , then r ag*= g; so m =g/g*--<g. If k > l , then

m = (2g - 1 + k ) / ( 2 g * - 1 + k)

-< (2g - 1 + k)/(k - 1)

- < 2 g + 1. [ ]

Remark. The maximum value m = 2g + 1 occurs only when k = 2. Otherwise m -< g + 1. If m = g + 1, then k --< 3. For all o ther values of k -> 4, m -< g.

C O R O L L A R Y 2.4. (cf. [15]). I l K is a libered knot in S 3 of genus g and period m, then m - < 2 g + l .

Proof. It is well known that a fibered knot has a unique incompressible Seifert surface, up to isotopy. See L e m m a 5.1. By Corollary 2.2 K has an invariant Seifert surface of genus g. By L e m m a 2.3 m-< 2g + 1. [ ]

See Section 6 for more precise results in this direction, where it will be shown that very few fibered knots of genus g admit periods greater than g.

3. Periodic knots have Property R

Let K be a knot in S 3 which is invariant under a semifree, or ientat ion- preserving action of a cyclic group Cm, with nonempty fixed point set A disjoint f rom K. The manifold obta ined by 0-surgery on K is defined to be M(K, O)= ( S 3 - i n t N ( K ) ) U B 2 • ~ where N ( K ) is a tubular ne ighborhood of K and the image of the meridian OB2• pt is a longitude of K (nul lhomologous in S 3 - K).

T H E O R E M 3.1. I l K is a periodic knot in S 3 and ~q(M(K, 0))~7/ , then K is unknoned (and M(K, 0) ~ S 1 x $2).

Proof. O n e may assume that the tubular ne ighborhood N ( K ) is invariant under a given C,, action. As in the proof of Theorem 2.1 K has an invariant

534 A L L A N L. E D M O N D S AND C H A R L E S LIVINGSTON

Seifert surface. Therefore the group action leaves a preferred longitude invariant, and it is possible to extend the action on S 3 - i n t N ( K ) to M(K,O). The fixed point set of the extended action is A U(0X $1). The quotient of M(K, O) by the induced Cm action may be described as the result of 0-surgery on the quotient of K under the original action. Put succinctly, M(K, 0)*= M(K*, 0). In particular then HI(M(K, 0)*; 7/)-~7/, generated by the image of the core 0 x S 1. Therefore M(K, 0)* contains a nonseparating incompressible surface S* (in fact a 2-sphere). It may be assumed that among all incompressible surfaces disk equivalent to S*, S* meets the branch set transversely in M(K, 0)* a minimum number of points.

By Theorem 2.1, and its proof, the inverse image S of S ' i n M(K, O) is an invariant, incompressible surface in M(K, 0). Because Irl(M(K, 0)) ~-7/, S must be a 2-sphere or a collection of m 2-spheres.

If S is not connected, then M(K, 0 ) - S consists of m components cyclically permuted by the action. In this case the action would have empty fixed point set. This contradiction implies that S is a single invariant 2-sphere, which must therefore meet the fixed point set in exactly two points.

Since the core 0 x S 1 of B 2 x S 1 represents a generator of H~(M(K, 0)), the intersection number of 0 • 1 with S must be +1. But S meets 0 x S 1 at most twice. Therefore S meets 0 x S 1 once and the original axis A once.

Finally, the manifold obtained by removing f rom M(K, 0) a small tubular neighborhood of 0 • S 1 is just S 3 - int N(K). But in this space S becomes a disk with boundary a longitude of K. Hence K is unknotted. [ ]

4. A c t i o n s on F 2 x [0, 1]

As the second step toward standardizing actions on fibered knots and other fibered 3-manifolds we show how to straighten actions on the product of a surface with the unit interval.

Let F be a compact surface and G be a finite group of diffeomorphisms of F • 1], leaving Fx{0} and Fx{1} invariant. The restriction of G to F• induces an action of G on F : g ( x ) = ~rFg(x, 0 ) w h e r e ~'v : F • 1]--~F is the projection. The associated straight action of G on F x [ 0 , 1] is given by g(x, t )=

(g(x), t).

T H E O R E M 4.1. Let F be a compact, orientable surface and let G be a finite group acting smoothly and effectively on F x[0, 1], preserving orientation and leaving F• and Fx{1} invariant. Assume that the orbit space FIG is not a 2-sphere with less than 4 branch points. Then the given action is equivalent to its associated straight action, by a diffeomorphism of F x [0, 1] which is the identity on Fx{O}.

Group actions on fibered three-manifolds 535

Proof. It is easy to see that one may assume F is connected: Separate orbits of components may clearly be handled separately; if F1 is component of F such that G(FI) = F, let H be the subgroup of G leaving F1 x[0, 1] invariant and suppose the action of H can be straightened, so there is a diffeomorphism 4 ~ : F l x [0, 1]--* FI x[0, 1] such that 4~ I F~ x{0} is the identity and ek(h(x), t) = hrb(x, t) for h e H, x �9 F~, t e [0, 1]; define 0 : F x [0, 1] ~ F x [0, 1] by 0(g(x), t) = g~b(x, t) for g c G, x �9 F~, t �9 [0, 1]; then 0 is the required equivariant diffeomorphism satisfy- ing 0(g(x), t) = gO(x, t) for all g �9 G, x �9 F, t �9 [0, 1].

So suppose F is connected. We proceed by induction on the ordered pairs (r, b) = (genus of F, number of components of aF), ordered lexicographically.

If r = 0, then b -> 1 since (0, 0) is an excluded case. If b = 1, then F is a disk. In this case G must be cyclic. The solution of the Smith Conjecture [19] then shows that the action on F x [ 0 , 1] can be straightened. Note that if one assumes the action is already straight on a F x [ 0 , 1], then the action can be straightened

tel OF x [0, 1]. Inductively, first consider the cases where F has nonempty boundary (b -> 1).

Covering space theory shows that we may assume the action is already straight on aF x [0, 1]. We show inductively that the action on F x [0, 1] can be straightened by a diffeomorphism which is the identity on F x{0}UOFx[0 , 1]. Choose an invariant family of disjoint, properly embedded arcs A1 . . . . . A,~ c F, where rn is the order of G, such that if F is cut open along A1,. �9 -, Am, then the complexity (r, b) is reduced. Such a family is easily constructed as the preimage of an arc A* in the orbit surface FIG which, if homotopic into OF~G, does not cut off a disk

containing less than two branch points. Let B~=Aix{1} in Fx{1},l<--i<--rn. This family of arcs is probably not

G-invariant. We may isotope {Bi}, rel end points, to an invariant collection as follows: Impose a G-invariant hyperbolic metric on F x {1} with totally geodesic boundary curves. (If S is an annulus, one must use a euclidean metric instead.) Each B~ is then homotopic, rel end points, to a unique geodesic arc C~ with the same end points as B~. By the uniqueness of the choice of these geodesics the

collection {C~} is G-invariant. Moreover {G} consists of embedded, pairwise disjoint arcs. The essential facts

here are that the minimum number of self-intersections in an arc representing a given relative homotopy class is realized by its geodesic representative and that the minimum number of intersections between two arcs representing two given relative homotopy classes is also realized by their geodesic representatives. A proof of these assertions follows the lines of the proof of the analogue of the second statement in the context of closed curves as given in [3; Expoge 3, Proposition 10]. Since the B~ are disjoint and embedded, it follows that the same

is true of the C~.

536 A L L A N L. E D M O N D S A N D C H A R L E S LIVINGSTON

The closed curves ~q = Ai • {0} U OA~ x [0, 1] U Ci form a G-invariant family of pairwise disjoint simple loops in 0(F• 1]). Each oq is nullhomotopic in F x [0, 1], since oq---A~ • {0} U 0A~ x [0, 1] U Ai x {1}. By the Equivariant Dehn Lemma [13], there is a G-invariant family of disjoint, embedded disks D~ c F x [0, 1] with 0D~ =oq. Since F x [ 0 , 1] is irreducible, U Di is isotopic to U A~ • 1]. There- fore, cutting F • 1] open along U D~ results in a new G-manifold of the form F ' x [0, 1] where the product structure imposed on the copies of Di extends that on-OF x [0, 1]. One may apply covering space theory to straighten the action on the copies of Di prior to cutting open along U D~. Now the inductive hypothesis says that the action on F ' x[0, 1] is equivalent rel OF' • 1] to the associated straight action. Gluing F ' • 1] back together along the cuts provides an equivalence of the given action on F x [0, 1] with its associated straight action.

Finally suppose aF = O and either the orbit space F* has positive genus or is a sphere with at least four branch points. Choose an invariant family of disjoint embedded simple loops Aa . . . . . A,~ c F, m--<[G[, which are homotopically non- trivial in F. Such a family is constructed as the preimage of a suitable closed loop in F* which does not bound a disk containing less than two branch points.

Let B~ = A~ • {1}, 1--<i--- < m. This family of loops is not in general G-invariant. But by replacing them with a corresponding set of geodesics for a G-invariant hyperbolic metric on F • {1} we obtain an invariant family of simple, closed loops {G} such that C~ = Bi for each i.

Now the Equivariant Dehn Lemma (for planar domains) of Meeks-Yau [13] implies that there is a G-invariant family of disjoint embedded annuli V~ F • 1] with 0V~ =A~ x{0}UC~. Since C~ =A~ x{1} and F x [ 0 , 1] is irreducible, V~ is isotopic rel A~ • 0 to Ai x [0, 1]. Thus the given action is equivalent to one which preserves the family {A~ • [0, 1]} and is straight there. Now cut open along U A~ • [0, 1], apply the inductive hypothesis, and glue back together to complete the argument. []

Remark. We have been informed that W. Meeks and G. P. Scott have proved Theorem 4.1 without the hypothesis that F/G is not a 2-sphere with fewer than four branch points. The proofs involve singular incompressible surfaces.

5. A c t i o n s on b u n ~ e s o v e r the c i rde

Let M denote a compact, connected, orientable 3-manitold which fibers over the circle S a, in such a way that aM, if nonempty, is a torus which inherits an induced fibering by restriction, and satisfies Ha(M; Q ) ~ Q . For example M might

Group actions on fibered three-manifolds 537

be the complement of an open tubular neighborhood of a fibered knot in a homology sphere. The condition on homology is used to insure that M admits an essentially unique fibering. The following lemma is well known.

LEMMA 5.1. Any two connected, properly embedded, non-separating, two- sided, incompressible surfaces F1, IrE : M are isotopic.

Proof sketch. One may suppose that F 1 is the fiber of a fibering M ~ S 1. Let ~t---F1 • be the corresponding infinite cyclic covering. Because F2 in non- separating, connected, and two-sided, intersection with /72 defines a surjective homomorphism Try(M) --~ Z, which must coincide, up to sign, with the correspond- ing homomorphism associated with the fibration since Hi(M; 7I)/torsion ~ 7. Thus /72 lifts homeomorphically to an incompressible surface ff2c/~7/. If lrl(P2)--> 7rl(/V/) were not also surjective, then an argument with Van Kampen's theorem would imply that ~rx(/V/) is not finitely generated. (Compare [20; p. 97].)

T H E O R E M 5.2. Let G be a finite group acting by orientation-preserving diffeomorphisms on the fibered 3-manifold M as above, with exceptional set E disjoint from 3M. Assume (i) that the orbit manifold M* is not S 2 x S ~ where S 2 x point meets the branch set in less than four points, and (ii) that Hi(M; Q)c Q. Then the given fibering of M is isotopic to a fibering in which G maps fibers into

fibers.

Proof. The quotient M* is a manifold with OM*, if nonempty, a torus. If O M ~ , this immediately implies H I ( M * ; Q ) ~ 0 . In general H : ( M * ; Q ) ~ Hi(M; ~)G, nonzero by assumption. Therefore M* contains a nonseparating, connected, two-sided incompressible surface F*. We may assume that F* meets the branch set B transversely in a minimal number of points. Then by Theorem 2.1 the preimage F of F* in M is an incompressible surface. Let F~ , . . . , / 7 , be the components of F. By Lemma 5.1 each component F~ is isotopic to a fiber of M. Cutting M open along F one obtains a group action on F • I. By Theorem 4.1 this action is equivalent to a straight action on F x I, each component of F x {t} being a fiber. Reidentifying F• with F• one obtains the required equivariant

fibering. [ ]

Remarks. If the action has fixed points then F is connected, and each fiber is G-invariant. The recent result of Meeks-Scott mentioned earlier shows that the hypothesis (i) above can be dropped.

C O R O L L A R Y 5.3. The quotient M* fibers over S ~. []

5 3 8 A L L A N L. E D M O N D S A N D C H A R L E S L I V I N G S T O N

I S F 3 /

Figure 1

An alternative e lementary proof of the corollary invokes Stallings' fibering theorem [20]. Compare [16].

6. Applications to fibered knots

Let K c S 3 be a fibered knot invariant under a smooth, orientation-preserving, semifree action of the cyclic group C,, of order m, having fixed axis A, a knot disjoint f rom K. The solution of the Smith Conjecture [19] implies that A is unknotted, and hence that the orbit space is again S 3 with branch set B = A* also an unknot ted circle. The following simply interprets Theorem 5.2 in this setting.

P R O P O S I T I O N 6.1. The fibering of K is isotopic to a fibering preserved by the action of Cm so that the axis A is transverse to the fibers; thus the quotient knot K* inherits a fibering with all fibers transverse to the branch set B. []

Let F c S 3 denote a typical fiber for the knot K transverse to A and let F* denote its image in the orbit space. Note that since the axis is connected the local two-dimensional representations of Cm in the tangent space of F at the points of F A A are all equivalent to rotation by +21rk/m for some fixed integer k.

Our first application of Proposition 6.1 is based on combining Corollary 2.4 with the fact that a genus 0 fibered knot is unknotted.

P R O P O S I T I O N 6.2. For g>--1 the only fibered knot of genus g which is invariant under an action of Cm as above, with m = 2g + 1, is the (2, 2g + 1) torus knot, up to orientation.

Proof. One can check using the Riemann-Hurwitz formula that Cm, m = 2g + 1, acts semifreely on a surface F of genus g with one boundary component,

Group actions on fibered three-manifolds 539

Figure 2

only so that the quotient F* is disk with two branch points. Therefore the quotient of an m-periodic knot K is an unknotted circle K* in S3, and the branch

set B is transverse to each disk fiber for K* , meeting each fiber in exactly two points. The only such link {K*, B} up to orientation is shown in Figure 2. One easily checks that the preimage of K* must be the (2, +m) torus knot. [ ]

PROPOSITION 6.3. For g >--1 the only fibered knots o f genus g which are

invariant under an action o f C,, as above, with m = g + 1, are the (3, ~ m ) torus

knot and the (3, + m ) Turk ' s head knot.

Pro@ The group C,, must act on the typical fiber F of genus g = m - 1 with orbit surface a disk and exactly three branch points. According to [10] the only three-stranded braids (up to conjugation in the braid group, and reversal of orientation) closing to unknotted circles, are the two depicted in Figure 3. The preimages of K* in the two cases are precisely, the (3, =~m) torus knot and the

(3, • Turk's head knot. [ ]

1<*

Figure 3

540 A L L A N L. E D M O N D S A N D C H A R L E S L I V I N G S T O N

R e m a r k . In working with specific examples the following observation, based on a fundamental result of Murasugi [15], is often helpful. If a knot K in S 3 is invariant under an action of the cyclic group C,, and the linking number of the axis with K is A, then A is completely determined by m. In the case of fibered knots this readily implies that the genus of the quotient fibered knot K* is determined by m as well.

To see this assertion about A, let p be any prime divisor of m, of order r in m. According to Murasugi

AK(t) m [8~ (t)]P'- ' [f( t)] ~' mod p,

where ~ x ( t ) = ( t x - 1 ) / ( t - 1 ) . Notice that 8~(t) has no repeated factors as a polynomial with coefficients in 2~p. This is because the derivative of t x - 1 has no nontrivial roots, since A is relatively pr ime to p.

If there were two Cm actions on (S 3, K) with axes having different linking numbers with K, say A1 and h2, then we would have

[Sx,(t)]o'-l[f,(t)] p" ~ [6x,(t)] p'-1 [/2(t)] p' mod p.

Now suppose g(t) is an irreducible factor of this product, of order n. Then one has

n = el(p" - 1) + ~ lp ' = e2(p' - 1) + SaP r,

where ei is the order of g in ~ and 8i is the order of g in/~. Since 8~ has no repeated roots, e~ is 0 or 1. One concludes that el = e2, and hence that the factorization of 8x, is the same as that of 8x:. In particular they have the same

degree, so h l = h2. The preceding uniqueness results do not hold in the presence of larger

numbers of branch points, as the following example (inspired by a similar construction of Morton [14]) shows.

E X A M P L E . There exist infinitely many distinct genus two fibered knots with (72 actions. Consider the link {K*, B,} in Figure 4, where n denotes the number of half twists. For all n, B , is unknotted. Let K~ be the preimage of K* in S 3 under the two-fold cover of S 3 branched along B,. Then K , is a genus two fibered knot with a C2 action.

The Alexander polynomial of K~ can be readily computed to be

Ar~(t) = # + (n2+ n -- 1 ) t 3 + ( - - 2 n 2 - 2 n + 1)t2+ (n2+ n - 1) t+ 1.

Thus the knots /4~ are all distinct, for n > 0.

( 541

K

Group actions on fibered three-manifolds

Figure 4

To compute AK.(t) proceed as follows: S 3 - K * is fibered by disks and hence S 3 - ( K * L A B . ) is fibered by 5 times punctured disks. The monodromy for that fibration is the product of twists o'i with each o-~ corresponding to one half twist in the braid B,. The monodromy for the genus 2 fibration of K , is the product of the lifts of the ~r~ to the branched covering space. The Alexander polynomial of K, is the characteristic polynomial of the monodromy. Details of the calculation are left

to the reader. In cases where the quotient knot has positive genus further complexities arise.

EXAMPLE. Consider the two unknotted curves B~ and B 2 in the comple- ment of the trefoil knot K * as depicted in Figure 5. In each case it is easy to check that Bi can be made transverse to the standard fibration of the trefoil complement, meeting each fiber exactly once. The preimage K1 of K* under the two-fold cover of S 3 branched along B1 is the granny knot. The preimage/<2 of K* under the two-fold cover of S 3 branched along B2 is the knot 821 in the

standard knot tables [17]. In [9; 1.28] D. Goldsmith posed the following question. Suppose p : M ~ S 3 is

a cyclic cover of degree m branched over a link B; suppose K * is an unknotted

f

Figure 5

K s

)

542 A L L A N L. E D M O N D S A N D C H A R L E S L M N G S T O N

s i m p l e c l o s e d c u r v e s u c h t h a t K = p - ~ ( D * ) is a f i b e r e d k n o t in M. Is B i s o t o p i c t o

a b r a i d a b o u t K * ? ( T h e s a m e q u e s t i o n m a k e s s e n s e if K * is j u s t a f i b e r e d k n o t . )

W h e n M is a r a t i o n a l h o m o l o g y s p h e r e , th i s is a n i m m e d i a t e c o n s e q u e n c e of

T h e o r e m 5.2. I f t h e d e g r e e m is a p r i m e p o w e r a n d B is c o n n e c t e d , t h e n M is

n e c e s s a r i l y a r a t i o n a l h o m o l o g y s p h e r e ( see [2; w so t h e a n s w e r is a l so yes in

th i s case .

REFERENCES

[1] BURDE, G., Uber periodische Knoten, Arch. Math. 30 (1978), 487-492. [2] CASSON, A. J. and GORDON, C. McA., On slice knots in dimension three, Proc. Amer. Math. Soc.

Summer Syrup. on Algebraic and Geometric Topology Stanford, 1976, pp. 39-54. [3] FATHI, A., LAUDENBACH, F. and POENARU, V., Travaux de Thruston sur les surfaces, Ast6risque

66-67 (1979). [4] GORDON, C. McA. and LrrrmRLA~, R. A., Incompressible surfaces in branched coverings, to

appear in: The Smith Conjecture, Proc. of a Conference at Columbia University, New York, 1979.

[5] , - - and MURASUGX, K., Signatures of covering links, Can. J. Math 33 (1981), 381-394. [6] HmLMAN, J. A., New proofs of two theorems on periodic knots, Arch. Math 37 (1981), 457-461. [7] - - , A survey of some results on symmetries and group actions on knots and links. Preprint. [8] KJM, P. K. and TOLLEFSON, J. L., PL involutions of fibered 3-manifolds, Trans. Amer. Math. Soc.

232 (1977), 221-237. [9] KIRaV, R., Problems in low dimensional manifold theory, Proc. Amer. Math. Soc. Summer Symp.

on Algebraic and Geometric Topology, Stanford, 1976, pp. 273-312. [10] MAGNUS, W. and PELUSO, A., On knot groups, Comm. Pure and App. Math. 20 (1967), 749-770. [11] LUDICKE, Ulrich, Zyklische Knoten, Arch. Math. 32 (1979), 588-599. [12] MEEKS, W. H. III and YAU, S. T., Topology of three dimensional manifolds and the embedding

problems in minimal surface theory, Annals of Math. 112 (1980), 441-485. [13] - - , The equivariant Dehn's lemma and loop theorem, Comm. Math. Helv. 56 (1981),

225-239. [14] MORTON, H. R., Infinitely many [ibered knots having the same Alexander Polynomial, Topology

17 (1978), 101-104. [15] MURASUGI, K., On per/od/c knots, Comm. Math. Helv. 4 (1971), 162-174. [16] - - , On Symmetry of Knots, Tsukuba J. Math 4 (1980), 331-247. [17] ROLFSEN, F., Knots and Links, Publish or Perish, Inc., Math. Lecture Series 7 (1976). [18] SAKUMA, M., Periods of Composite Links, Math. Seminar. Notes. Kobe Univ. 9 (1981). [19] The Smith Conjecture, Proc. of a Conference at Columbia University, New York, 1979, to

appear. [20] STAtaJNOS, J., On fibering certain 3-mainfolds, Topo/ogy of 3-mani[olds and Related Topics, M.

K. Fort, Editor, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. [21] ToN ~psor~, J. L , Periodic homeomorphisms of 3-manifolds fibered over S 1, Trans. Amer. Math.

Sot:. 123 (1976), 223-234. Erratum, ibid. 243 (1978), 309-310. [22] TROTTER, H. F., Periodic automorphisms of groups and knots, Duke Math. J. 28 (1961), 553-557. [23] WALDHAUSEN, F., Irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968),

55-88.

Department of Mathematics Indiana University Bloomington, Indiana 47405

Received September 9, 1982